Definition

Consider a return time series $r_t=\mu +{\epsilon }_t$, where $\mu$ is the expected return and ${\epsilon }_t$ is a zero-mean white noise. Despite of being serially uncorrelated, the series ${\epsilon }_t$ does not need to be serially independent. For instance, it can present conditional heteroskedasticity. The Exponential $\mathrm{GARCH}$ $\left(\mathrm{EGARCH}\right)$ model assumes a specific parametric form for this conditional heteroskedasticity. More specifically, we say that ${\epsilon }_t~\mathrm{EGARCH}$ if we can write ${\epsilon }_t={\sigma }_t{z}_t$, where $z_t$ is standard Gaussian and:

$$\text{ln}\left({\sigma }_t^2\right)=\omega +\alpha \left(\left|z_{t-1}\right|-𝔼\left[\left|z_{t-1}\right|\right]\right)+\gamma z_{t-1}+\beta \text{ln}\left({\sigma }_{t-1}^2\right)$$

Estimation

V-Lab estimates all the parameters $\left(\mu ,\omega ,\alpha ,\gamma ,\beta \right)$ simultaneously, by maximizing the log likelihood. The assumption that $z_t$ is Gaussian does not imply the returns are Gaussian. Even though their conditional distribution is Gaussian, it can be proved that their unconditional distribution presents excess kurtosis (fat tails). In fact, assuming that the conditional distribution is Gaussian is not as restrictive as it seems: even if the true distribution is different, the so-called Quasi-Maximum Likelihood (QML) estimator is still consistent, under fairly mild regularity conditions.

Besides leptokurtic returns, the $\mathrm{EGARCH}$ model, as the $\mathrm{GARCH}$ model, captures other stylized facts in financial time series, like volatility clustering. The volatility is more likely to be high at time $t$ if it was also high at time $t-1$. Another way of seeing this is noting that a shock at time$t-1$ also impacts the variance at time $t$.

The $\mathrm{EGARCH}$ model does not require any restriction on the parameters because, since the equation is on log variance instead of variance itself, the positivity of the variance is automatically satisfied, and that is the main advantage of the $\mathrm{EGARCH}$ model. In general, the likelihood maximization with no restrictions results in faster and more reliable optimizations.

Prediction

Let $r_t$ be the last observation in the sample, and let $\hat{\omega }$, $\hat{\alpha }$, $\hat{\gamma }$, and $\hat{\beta }$ be the QML estimators of the parameters $\omega$, $\alpha$, $\gamma$ and $\beta$, respectively. When $z_t$ are i.i.d. Gaussian, we have:

$$𝔼\left[\text{exp}\left\{\alpha \left(\left|z_t\right|-𝔼\left(\left|z_t\right|\right)\right)+\gamma z_t\right\}\right]=\text{exp}\left(-\hat{\alpha }\sqrt{\frac{2}{\pi }}\right)\left[\text{exp}\left(\frac{{\left(\hat{\gamma }+\hat{\alpha }\right)}^2}{2}\right)\Phi \left(\hat{\gamma }+\hat{\alpha }\right)+\text{exp}\left(\frac{{\left(\hat{\gamma }-\hat{\alpha }\right)}^2}{2}\right)\Phi \left(\hat{\alpha }-\hat{\gamma }\right)\right]$$

The EGARCH model thus implies that the forecast of the conditional variance at time $T+h$, $h\ge 2$, is given by:

$$\begin{array}{lcr}{{\hat{\sigma }}_{T+h}}^2\hfill & \hfill =\hfill & 𝔼\left[{{\sigma }_{T+h}}^2|r_T\text{,}{r}_{T-1}\text{, ...}\right]\hfill \\ \hfill & \hfill =\hfill & {\left({{\sigma }_{T+1}}^2\right)}^{{\hat{\beta }}^{h-1}}\text{exp}\left\{\frac{1-{\hat{\beta }}^{h-1}}{1-\hat{\beta }}\left(\hat{\omega }-\hat{\alpha }\sqrt{\frac{2}{\pi }}\right)\right\}⨯\hfill \\ \hfill & \hfill \hfill & \hfill \underset{i=0}{\overset{h-2}{\Pi }}\left[\text{exp}\left(\frac{{\left({\hat{\beta }}^i\hat{\gamma }+{\hat{\beta }}^i\hat{\alpha }\right)}^2}{2}\right)\Phi \left({\hat{\beta }}^i\hat{\gamma }+{\hat{\beta }}^i\hat{\alpha }\right)+\text{exp}\left(\frac{{\left({\hat{\beta }}^i\hat{\gamma }-{\hat{\beta }}^i\hat{\alpha }\right)}^2}{2}\right)\Phi \left({\hat{\beta }}^i\hat{\alpha }-{\hat{\beta }}^i\hat{\gamma }\right)\right]\end{array}$$

where $\Phi \left(\cdot \right)$ is the standard Gaussian CDF. The proof is easily achieved using the recursive formulation of the log-variance.

And so, by applying the above formula iteratively, we can forecast the conditional variance for any horizon $h$. Then, the forecast of the compound volatility at time $T+h$ is:

$${\hat{\sigma }}_{T+1:T+h}=\sqrt{\sum _{i=1}^h{\hat{\sigma }}_{T+i}^2}$$

EGARCH vs. GARCH

There is a stylized fact that the $\mathrm{EGARCH}$ model captures that is not contemplated by the $\mathrm{GARCH}$ model, which is the empirically observed fact that negative shocks at time $t-1$ have a stronger impact in the variance at time $t$ than positive shocks. This asymmetry used to be called leverage effect because the increase in risk was believed to come from the increased leverage induced by a negative shock, but nowadays we know that this channel is just too small. Notice that the effective coefficient associated with a negative shock is $\gamma -\alpha$, while the effective coefficient associated with a positive shock is $\gamma +\alpha$. In financial time series, we generally find that $\gamma$ is negative and statistically significant.

EGARCH(p,q)

The specific model just described can be generalized to account for more lags in the conditional variance. An $\mathrm{EGARCH}\left(p,q\right)$ model assumes that:

$$\text{ln}\left({\sigma }_t^2\right)=\omega +\sum _{i=1}^p\left\{{\alpha }_i\left(\left|z_{t-i}\right|-𝔼\left[\left|z_{t-i}\right|\right]\right)+{\gamma }_i{z}_{t-i}\right\}+\sum _{j=1}^q{\beta }_j\text{ln}\left({\sigma }_{t-j}^2\right)$$

The best model ($p$ and $q$) can be chosen, for instance, by Bayesian Information Criterion (BIC), also known as Schwarz Information Criterion (SIC), or by Akaike Information Criterion (AIC). The former tends to be more parsimonious than the latter. V-Lab uses $p=1$ and $q=1$, because this is usually the option that best fits financial time series.